dc.title: Real-time inverse solutions for digital twins dc.description.abstract: Digital twins require rapid data assimilation for digital state updating and downstream tasks, such as prediction and control. For many physical systems, the data assimilation task requires the solution of a partial differential equation (PDE)-constrained inverse problem, which is often computationally intractable in real time using traditional PDE solvers. This work develops computational methods to enable the real-time, online solution of inverse problems for the digital twin setting.
To achieve real-time performance, the methods developed in this thesis seek to exploit known structure in the PDE to enable rapid computation of inverse solutions. Specifically, for linear settings, the inverse problem can be posed as a regularized least-squares problem with a linear constraint arising from the governing PDE. As a result of this formulation and problem structure, a closed-form solution is given by the normal equations. This closed-form expression admits an offline-online decomposition, where offline pre-computation of the high-dimensional matrix operations in the resulting inverse map enables rapid online evaluation of the unknown parameters and corresponding integrated quantities of interest. This inverse solution captures the PDE-governed physics behavior, while achieving the desired goals of fast online speeds with quantifiable uncertainty.
In the nonlinear case, scientific machine learning (SciML) methods are used to pre-train models that accelerate the inverse solutions online. First, a direct inverse mapping from the observables to the unknown inversion parameters is learned via SciML. In this work, optimal classification trees (OCTs) are proposed as an interpretable, rapid-to-evaluate inverse mapping. The OCT is trained offline using high-fidelity physics simulation data, and evaluation of the learned OCT provides real-time inverse solutions online. Second, a reduced-basis neural operator approach is presented that specifically targets the class of problems with spatiotemporal dynamics governed by PDEs that are parameterized nonlinearly with respect to model parameters, and linearly with respect to inversion parameters. Based on this physics structure, the neural operator approximates the nonlinear map from the model parameters to the parameter-to-observable operator in a reduced subspace. Since the output of the neural operator is the parameter-to-observable operator itself (manifested as a matrix), the approach is referred to as NEural Matrix Operator (NEMO). With NEMO, for given model parameters, a closed-form inverse problem solution in a reduced subspace is available, analogous to the linear setting, with rapid evaluation speeds and uncertainty quantification.
The methods are demonstrated on several engineering applications, including airborne contaminant initial condition identification, and aerodynamic pressure load estimation for hypersonics, with a focus on the latter. The numerical studies in this work demonstrate high quality inverse problem solutions, obtained with several orders of magnitude online speedup compared to traditional PDE methods for digital twin data assimilation in real time.